∫2x5+x2dx5+x2=t2xdx=dt∫dtt=∫t−12dt=t−12+1−12+1+c=2t+c=25+x2+c

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$\int \frac{2 x}{\sqrt{5 + x^{2}}} d x =$

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$2 \sqrt{5 - x^{2}} + c$

$\log (5 + x^{2}) + c$

$2 \sqrt{5 + x^{2}} + k$

$x + c$

answer is A.

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Detailed Solution

$\int \frac{2 x}{\sqrt{5 + x^{2}}} d x$

$5 + x^{2} = t$

$2 x d x = d t$

$\int \frac{d t}{\sqrt{t}}$

$= \int t^{- \frac{1}{2}} d t$

$= \frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + c$

$= 2 \sqrt{t} + c$

$= 2 \sqrt{5 + x^{2}} + c$

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